JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2103, doi:10.1029/2002JB002082, 2003
Self-consistency in reference frames, geocenter definition, and surface
loading of the solid Earth
Geoffrey Blewitt1
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Nevada, USA
Received 8 July 2002; revised 30 October 2002; accepted 18 November 2002; published 19 February 2003.
[1] Crustal motion can be described as a vector displacement field, which depends on
both the physical deformation and the reference frame. Self-consistent descriptions of
surface kinematics must account for the dynamic relationship between the Earth’s surface
and the frame origin at some defined center of the Earth, which is governed by the Earth’s
response to the degree-one spherical harmonic component of surface loads. Terrestrial
reference frames are defined here as ‘‘isomorphic’’ if the computed surface displacements
functionally accord with load Love number theory. Isomorphic frames are shown to move
relative to each other along the direction of the load’s center of mass. The following
frames are isomorphic: center of mass of the solid Earth, center of mass of the entire Earth
system, no-net translation of the surface, no-net horizontal translation of the surface, and
no-net vertical translation of the surface. The theory predicts different degree-one load
Love numbers and geocenter motion for specific isomorphic frames. Under a change in
center of mass of surface load in any isomorphic frame, the total surface displacement
field consists not only of a geocenter translation in inertial space, but must also be
accompanied by surface deformation. Therefore estimation of geocenter displacement
should account for this deformation. Even very long baseline interferometry (VLBI) is
sensitive to geocenter displacement, as the accompanying deformation changes baseline
lengths. The choice of specific isomorphic frame can facilitate scientific interpretation; the
theory presented here clarifies how coordinate displacements and horizontal versus
INDEX TERMS: 1229 Geodesy and Gravity:
vertical motion are critically tied to this choice.
Reference systems; 1247 Geodesy and Gravity: Terrestrial reference systems; 1213 Geodesy and Gravity:
Earth’s interior—dynamics (8115, 8120); 1223 Geodesy and Gravity: Ocean/Earth/atmosphere interactions
(3339); KEYWORDS: reference frame, geocenter, surface loading, deformation, solid Earth, isomorphic
Citation: Blewitt, G., Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth,
J. Geophys. Res., 108(B2), 2103, doi:10.1029/2002JB002082, 2003.
1. Introduction
[2] Terrestrial reference frames play an integral role in the
interpretation of global-scale, low-frequency phenomena in
space-geodetic data [Argus et al., 1999; Chao et al., 1987;
Trupin et al., 1992; Dong et al., 1997]. Indeed, the degree to
which we can separate different processes in the Earth
system using geodesy depends critically on our ability to
realize accurate, stable reference frames, such as the International Earth Rotation Service (IERS) Terrestrial Reference
Frame (ITRF) [Altamimi et al., 2001]. For example, global
sea level rise depends directly on the global balance of
hydrologic processes and indirectly on crustal deformation
due to various processes such as postglacial rebound and
loading associated with the global hydrologic system itself.
This situation requires a terrestrial reference frame defined
1
Also at School of Civil Engineering and Geosciences, University of
Newcastle upon Tyne, UK.
Copyright 2003 by the American Geophysical Union.
0148-0227/03/2002JB002082$09.00
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consistently with the dynamics of loading so that crustal
deformation can be accurately related to its effect on sea
level. Global-scale processes typically have competing
models that differ at the millimeter level, yet the differences
at this level can have profound scientific implications and
can lead to broad variation in socio-economic impact
[Bilham, 1991].
[3] Reference frame realization and the separation of
processes are coupled problems. For example, conservation
of momentum demands that the center of mass of the Earth
system (CM) is a kinematic fixed point, invariant to
terrestrial dynamic processes. Realization of a frame centered on CM therefore plays a role in assessing the integration of various mass transport models and in defining
‘‘vertical displacement’’ for problems such as sea level rise.
While CM can be realized through the well-modeled
dynamics of Satellite Laser Ranging (SLR) satellites [Watkins and Eanes, 1997; Chen et al., 1999], one problem is
spatial and temporal aliasing, in that the sparsely observed
SLR network is continuously deforming by various loading
phenomena [Van Dam et al., 2001]. The Global Positioning
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System (GPS) has the converse problem. GPS satellites are
subject to complex nongravitational dynamics, which limits
the dynamic interpretation of station motions in the CM
frame in terms of global mass redistribution. However, GPS
provides the level of high spatial and temporal resolution
that is desirable for the study of solid Earth processes
[Blewitt, 1993]. Therefore it is not sufficient to rely entirely
on the CM frame for interpretation, but rather it is imperative to develop a theoretical framework that connects the
CM frame to other, geometrically defined frames that are
more generally accessible to techniques such as GPS and
very long baseline interferometry (VLBI).
[4] Here I aim to develop self-consistency in dynamics
models and terrestrial reference frame theory, as a contribution to improving geodetic practice and scientific interpretation. An important objective is to define terrestrial
reference frames that are compatible with dynamic theory.
Toward this objective I approach the general theoretical
problem of describing vector displacements of the Earth’s
surface when the origin of the reference frame is dynamically related to surface loading. Degree-one deformation is
fundamental to this problem [Blewitt et al., 2001]. Another
objective is to develop a methodology to classify frames,
and so provide some indication as to which frame might be
best suited to the research question at hand. For example, in
one extreme case that requires explanation in this paper,
Blewitt et al. [2001, p.2343] remarked that surface displacements associated with a change in the load’s center of mass
could be equally characterized as ‘‘either purely vertical or
purely horizontal’’ depending on the selection of reference
frame. Toward this objective I develop the concept of a
general class of ‘‘isomorphic frames.’’ Such frames provide
dynamically consistent kinematic descriptions of Earth
deformation despite predicting very different station coordinate time series. The apparent paradox will be explained
in depth.
[5] The theoretical framework developed here should be
useful (1) for comparing published results derived using
different types of frames, (2) for selecting a frame appropriate to the problem at hand, (3) for self-consistent use of
reference frames, data, and models, and (4) for the integration of space geodetic techniques. This work builds on the
theory of geocenter motions of Trupin et al. [1992] and
Dong et al. [1997], on reference frame concepts of Argus
[1996], Heki [1996], and Argus et al. [1999], on the spectral
loading theory of Mitrovica et al. [1994], and on degree-one
deformation concepts of Blewitt et al. [2001]. Figure 1
presents a contextual overview of the work presented here,
showing how these developments link to dynamic theory
and to computed observation models.
2. Solid Earth Loading Dynamics
2.1. Model Considerations
[6] Loading models have traditionally used Green’s functions, as derived by Farrell [1972] and applied in various
geodetic investigations [e.g., Van Dam et al., 1994]. The
Green’s function approach is fundamentally based on load
Love number theory, in which the Earth’s deformation
response is a function of the spherical harmonic components
of the incremental gravitational potential created by the
surface load. To study the interaction between loading
Figure 1. The basic elements of our analytical loading
model, identifying how self-consistency between loading
dynamics and the reference frame can be incorporated.
Phenomena are in round boxes, measurement types are in
rectangular boxes, and physical principles are attached to
the connecting arrows. The arrows indicate the direction
leading toward the computation of measurement models.
dynamics and the terrestrial reference frame, it is convenient
to use the spherical harmonic approach [Lambeck, 1988;
Mitrovica et al., 1994; Grafarend et al., 1997] (therefore the
conclusions would also apply to the use of Green’s functions). The following ‘‘standard model’’ is based on a
spherically symmetric, radially layered, elastic Earth statically loaded by a thin shell on the Earth’s surface. Farrell
[1972] used such a model to derive Green’s functions that
are now prevalent in atmospheric and hydrological loading
models [e.g., Van Dam et al., 2001]. The preliminary
reference Earth model (PREM) [Dziewonski and Anderson,
1981] yields load Love numbers almost identical to those of
Farrell [Lambeck, 1988; Grafarend et al., 1997].
[7] Whereas it is in widespread use, the standard model
might be improved by incorporating the Earth’s ellipticity
[Wahr, 1981], mantle heterogeneity [Dziewonski et al.,
1977; Su et al., 1994; Van Dam et al., 1994; Plag et al.,
1996], and Maxwell rheology [Peltier, 1974; Lambeck,
1988; Mitrovica et al., 1994]. There is no consensus model
to replace PREM yet; however, the general approach to
reference frame considerations described here would be
applicable to improved models (though it might require
numerical rather than closed form analysis). The key point
is it that a solid understanding on how to account for the
choice of frame when interpreting data is needed for all
models, especially those in routine use, and that a lack of
this understanding might lead to errors of interpretation.
2.2. Equilibrium Tide
[8] It is analytically convenient to decompose the Earth
system as a spherical, solid Earth of radius RE and surface
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BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
mass that is free to redistribute in a thin surface layer (RE)
of surface density s() which is a function of geographical
position (latitude j, longitude l). Let us express the total
redistributed load as a spherical harmonic expansion:
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surface due to loading. From equations (5) and (3), the body
potential is
U 0 ðÞ ¼ U ðÞ gsh ðÞ ¼
X
1 þ kn0 h0n Vn ðÞ:
ð6Þ
n
sðÞ ¼
1 X
n X
S
X
s
nm Ynm ðÞ;
ð1Þ
n¼1 m¼0 ¼C
where Ynm
() are defined in terms of associated Legendre
C
S
= Pnm (sin j) cos ml, Ynm
= Pnm (sin j)
polynomials Ynm
sin ml. The summation begins at degree n = 1 assuming
that mass is conserved in the Earth system. It is this initial
degree-one term that is used in section 3 to address the
reference frame problem.
[9] It can be shown [e.g., Bomford, 1980, p.408] that, for
a rigid Earth, such a thin-shell model produces the following incremental gravitational potential at the Earth’s surface,
which we call the ‘‘load potential’’:
V ðÞ ¼
X
Vn ðÞ ¼
4pR3E g
n
XXX
ME
n
m
s
nm Ynm ðÞ
ð2n þ 1Þ
;
ð2Þ
where g is acceleration due to gravity at the Earth’s surface
and ME is the mass of the Earth. This load potential results
in a displacement of the geoid, called the ‘‘equilibrium
tide.’’ As shall be addressed later, the load deforms the solid
Earth, and in doing so creates an additional potential.
2.3. Surface Deformation
[10] According to load Love number theory, solutions for
surface displacements sh() in the local height direction
and sl () in any lateral direction specified by unit vector
^l () are given [Lambeck, 1988] by
sh ðÞ ¼
sl ðÞ ¼
X
n
X
h0n Vn ðÞ=g
ln0 ^lðÞ. rVn ðÞ=g;
ð3Þ
n
and the additional potential caused by the resulting deformation is
V ðÞ ¼
X
kn0 Vn ðÞ;
ð4Þ
n
where h0n, l0n, and k0n are degree-n load Love numbers, with
the prime distinguishing Love numbers used in loading
theory from those used in tidal theory. The surface gradient
^ ð1= cos fÞ@l , where j
^ and
^ @f þ l
operator is defined r ¼ j
^ are unit vectors pointing northward and eastward,
l
respectively.
[11] The net loading potential (load plus additional potential) relative to an Eulerian observer (the ‘‘space potential’’
as observed on a geocentric reference surface) is
U ðÞ ¼ V ðÞ þ V ðÞ ¼
X
1 þ kn0 Vn ðÞ:
ð5Þ
n
The net loading potential relative to a Lagrangean observer
(the ‘‘body potential’’ as observed on the deforming Earth’s
surface) must also account for the lowering of Earth’s
Therefore the ‘‘space’’ and ‘‘body’’ combinations of load
Love number, (1 + k0n) and (1 + k0n h0n) are relevant to
computing gravity acting on Earth-orbiting satellites and
Earth-fixed instruments, respectively.
[12] Solutions for surface deformations of the thin-shell
loading model are found by substituting equation (2) into
equations (3) and (5):
sh ðÞ ¼
4pR3E X X X 0 s
Y ðÞ
hn nm nm
;
ME n m 2n þ 1
sl ðÞ ¼
^l. rYnm
4pR3E X X X 0 s
ðÞ
ln nm
;
ME n m 2n þ 1
U ðÞ ¼
ð7Þ
s Y ðÞ
4pR3E X X X 1 þ kn0 nm nm
:
ME n m 2n þ 1
3. Linking Dynamics to the Reference Frame
[13] Equation (7) tells us the difference in position (and
potential) before and after imposing a load surface density
distribution parameterized in terms of spherical harmonic
coefficients. It will now be shown that the degree-one
contributions depend on the choice of reference frame,
specifically how the origin moves relative to the deforming
Earth. A fundamental problem that will now be addressed is
that a constant translation can be applied to equation (7)
without changing its form. Note that a constant translation is
a special case of a degree-one surface displacement field,
which alone would describe the degree-one loading response
only if the Earth were perfectly rigid. The degree-one
deformation field for a nonrigid Earth can be described as
a combination of deformation field plus a translation.
3.1. Degree-One Load Deformations
[14] For reference frame analysis it is useful to reformulate the degree-one potential in terms of the center of mass
of the load. From equation (2) we have
V1 ðÞ ¼
4pa3 g C C
S
C
s11 Y11 ðÞ þ sS11 Y11
ðÞ þ sC10 Y10
ðÞ ;
3ME
ð8Þ
C
S
C
where Y11
() = cosj cosl, Y11
() = cosj sinl, Y10
() =
sin j. Blewitt et al. [2001] encapsulate the relevant properties
of the load defining the load moment vector
m ¼ ML rL ;
ð9Þ
where rL is the change in the load’s center of mass (relative
to the solid Earth center of mass) and ML is the mass of that
portion of the load going into the calculation of rL . Typical
magnitudes for the case of seasonal groundwater variations
are rL 106 m and ML 1016 kg. Therefore, unlike the
case for the actual deformations (
10 mm), m is not
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BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
sensitive to the specific choice of reference frame, as long as
it is approximately geocentric. Note that m has the useful
property that it is independent of whether we reckon
stationary mass as part of the ‘‘redistributed’’ load.
[15] As the load is restricted to move on a spherical
surface of radius RE, equation (9) is equivalent to
ZZ
m¼
¼ R3E
^ðÞsðÞR2 d
RE h
E
ZZ
C
S
C
Y11 ðÞ^x þ Y11
ðÞ^y þ Y10
ðÞ^z sðÞd;
ð10Þ
where d = sinl dldj is an element of solid angle, ĥ() is
a unit vector pointing locally upward, and we use the
C
S
C
, ĥ.^y = Y11
, and ĥ.^z = Y10
. Substitution of
identities ĥ.^x = Y11
equation (1) into equation (10) and performing the
integration shows explicitly a one-to-one correspondence
between load moment components and degree-one coefficients of surface density:
4pR3E C S C mx ; my ; mz ¼
s11 ; s11 ; s10 :
3
ð11Þ
g C
S
C
^ðÞ. m=ME :
mx Y11
ðÞ þ my Y11
ðÞþmz Y10
ðÞ ¼ gh
ME
ð12Þ
[17] Blewitt et al. [2001] derived the same equation less
rigorously as a ‘‘first-order’’ calculation, arguing that the
total momentum of the Earth-load system is conserved and
that thus the solid Earth must experience a tide-raising
potential as it is displaced relative to CM through its own
degree-zero gravity field. The derivation here proves the
postulate of Blewitt et al. [2001] that equation (12) corresponds to the degree-one component of the load potential.
Substituting the vector form of equation (12) into equations
(3) and (5) gives us the following alternative form for the
degree-one deformations (where, to reduce clutter, the functional dependence on is now implicit):
^ m=ME ;
sh ¼ h01 V1 =g ¼ h01 h.
sl ¼ l10 ^l.rV1 =g ¼ l10 ^l. m=ME ;
^ m=ME :
U1 ¼ 1 þ k10 V1 ¼ 1 þ k10 gh.
½tB A ¼ ½aB A m=ME ;
ð14Þ
where the ‘‘isomorphic parameter’’ a depends on the
conceptual definition of the reference frame origin and ME
ensures that a is dimensionless. The subscripted square
brackets indicate to which frame the parameter refers. In the
new frame B the deformations appear to be
^ B ;
½sh B ¼ sh h.t
A
½sl B ¼ sl ^l.tB A ;
^ B :
½U B ¼ U gh.t
A
ð15Þ
Inserting equations (13) and (14) into equation (15) and
rearranging, we find
[16] Hence, equation (8) can be reformulated as
V1 ðÞ ¼
degree-one deformations take the form of equation (13).
What distinguishes specific isomorphic frames is the set of
load Love numbers; but the functional dependence on the
load moment is invariant.
[19] I now demonstrate that isomorphic frames have the
property that, under redistribution of the load characterized
by load moment m, the origins translate relative to each
along the axis of m. First let us postulate this and parameterize the translation of frame B with respect to frame A by
ð13Þ
The derivation of sl in this equation uses an identity in
vector analysis, proved in Appendix A. Next, I show that
load Love numbers depend on how the origin of the
reference frame translates relative to the Earth’s surface
when the load is redistributed. Specifically, the form of
equation (13) is independent under translations along the
direction of m.
3.2. Isomorphic Frame Transformations
[18] Let us assume we have a specified Earth model
reference frame in which load Love numbers have been
derived. How should we transform the load Love numbers
into different reference frames? In the context of loading
theory, I define a reference frame as ‘‘isomorphic’’ if the
^ m=ME ;
½sh B ¼ h01 aB A h.
0
½sl B ¼ l1 aB A^l. m=ME ;
^ m=ME :
½U B ¼ 1 þ k10 aB A gh.
ð16Þ
Therefore, equation (16) has the same form as equation
(13), where new degree-one load Love numbers are defined
by
0
h1 B ¼ h01 aB A ;
0
l1 B ¼ l10 aB A ;
1 þ k10 B ¼ 1 þ k10 aB A :
ð17Þ
Note that a translation of the frame origin only affects the
degree-one load Love numbers; therefore, equation (17)
does not apply to higher degrees. This can be understood if
we consider a rigid body translation as a special case of
degree-one deformation and consider that higher degree
deformations are orthogonal to degree-one; hence, the
degree-one components account for the translation entirely.
[20] It remains for us to calculate the isomorphic parameter a for the several types of reference frames used in global
deformation research and then use equation (17) to compute
the new load Love numbers. Once this is achieved, then
equation (7) can be used to compute deformations in the new
frame, given a model of the load. Alternatively, inversion
using geodetic data on surface displacements might be used
to estimate the load surface density distribution snm , for
which the load Love numbers must be transformed to the
geodetic frame of choice [Blewitt et al., 2001].
3.3. Reference Frame Properties of Degree-One
Deformations
[21] Equation (16) can be used to illustrate properties of
degree-one deformations that might be counterintuitive.
BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
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Figure 2. Degree-one deformations in various frames, generated by equation (16). A cross section of
the Earth is shown where the load moment points along the Earth’s axis toward the North Pole (top of the
figure). Filled circles show stations on the Earth’s surface (solid line) prior to deformation. Open circles
show stations on the Earth’s surface (dashed line) after deformation for each indicated frame. By
definition, each frame origin (+) remains stationary during mass redistribution and subsequent
deformation. Within each frame, CM is displaced from the origin to the location shown ().
Figure 2 was computer-generated assuming specific values
of a in equation (16), starting with Farrell’s load Love
numbers given by equation (18). Figure 2 illustrates an
exaggerated degree-one deformation field as seen in the
frames variously defined in the next section of this paper,
where the load moment points along the Earth’s axis toward
the North Pole (top of the figure). Starting with stations
(filled circles) on the Earth’s surface (solid line), the Earth’s
surface deforms (dashed line) and the stations move to their
final positions (open circles).
[22] The following key characteristics of degree-one
deformation are noted: (1) the Earth’s surface remains a
perfect sphere that is displaced with respect to the original
sphere; (2) the stations move closer together in the Northern
Hemisphere and further apart in the Southern Hemisphere;
(3) the shape of the final polyhedron of stations is frame
invariant; (4) the difference in sets of final station coordinates in each frame is given by a common translation (often
called the ‘‘geocenter’’); and therefore (5) in all frames the
difference between initial and final station coordinates can
be described by a frame-independent deformation plus a
frame-dependent translation.
[23] In addition to these characteristics, note that a new
frame can always be found for which any of the three load
Love numbers are zero. This implies that any degree-one
deformation field can be transformed into a frame in which
the displacements appear either purely vertical (the CL
frame in Figure 2) or purely horizontal (CH frame). Both
points of view are equally valid. There is no fundamental
distinction between the concepts of vertical motion versus
horizontal motion, as they depend entirely on the choice of
reference frame. Therefore, great care must be exercised
when interpreting global-scale deformations involving
global-scale redistribution of surface load, especially when
questions use the terms vertical and horizontal.
[24] Another unusual property is that if we chose a frame
in which the surface motion is entirely horizontal (CH
frame), then station motions actually peak at a maximal
distance (90) away from the point of maximum/minimum
loads. In such a frame, peak motions at the equator correspond to peak loads at the poles. This illustrates severe
limitations in nonglobal models of loading that use linear
regression of modeled displacement to local pressure, such
as recommended in the IERS conventions [McCarthy, 1996].
From a terrestrial reference frame perspective, local models
are not useful and are likely to be misleading (perhaps
improving scatter, but not accuracy) for purposes of separation of global-scale Earth processes.
4. Linking Frame Theory to Frame Practice
[25] There are several conceptually different types of
reference frame that have been used in theory and practice
[Heflin et al., 1992; Watkins et al., 1994; Van Dam et al.,
1994; Argus, 1996; Heki, 1996; Dong et al., 1997; Ma and
Ryan, 1997; Ray, 1998; Argus et al., 1999; Chen et al.,
1999; Gerasimenko and Kato, 2000; Altamimi et al., 2001].
Love number transformations given by equation (17) are
only applicable for the class of isomorphic reference frames
which, under a redistribution of load, move relative to each
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BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
other along the axis of the load moment, m. Fortunately,
many conceptual reference frames used in practice satisfy
this criterion, including frames for which the origin is
defined in terms of center of mass or some geometric center
of figure. We shall now consider specific conceptual isomorphic frames, which allow us to calculate then how the
origins and the degree-one load Love numbers transform
between them.
4.1. Center of Mass of the Solid Earth (CE)
[26] Let us first consider a reference frame fixed to the
center of mass of the solid Earth, CE. As shall be discussed
later, CE does change its trajectory in inertial space when
surface mass is redistributed. However, once the mass has
been redistributed to its final configuration, any resulting
deformation of the solid Earth cannot itself change the solid
Earth’s center of mass. Hence, the degree-one space potential of equation (5) remains invariant to deformation in the
CE frame, so [k10]CE = 0 Therefore, the CE frame is a natural
frame to compute the dynamics of solid Earth deformation
and to model load Love numbers. Farrell [1972] specifies
that in the CE frame
0
h1 CE ¼ 0:290;
0
l1 CE ¼ 0:113;
1 þ k10 CE ¼ 1:
ð18Þ
[27] A major disadvantage of the CE frame, however, is
that it is not directly accessible to observation. As pointed
out by Dong et al. [1997], the CE frame closely approximates the CF frame (center of surface figure, defined
below). However, to rigorously compare observation to
model requires a transformation of coordinates from the
model CE frame into the appropriate observational reference frame. The alternative given in this paper is to apply
equation (17) to transform the load Love numbers to an
appropriate observational frame.
4.2. Center of Mass of the Earth System (CM)
[28] The center of mass of the entire Earth system
includes the solid Earth and surface load. CM is stationary
with respect to satellite orbits in inertial space and is
therefore an appropriate frame for modeling SLR measurements. In practice, for GPS, the procedure of simultaneously
estimating satellite orbits and fiducial-free station positions
[Heflin et al., 1992] naturally produces station coordinates
in the CM frame. While the internal geometry of a fiducialfree network can be very precise, the external solution is
typically an order of magnitude less precise due to sensitivity of a global translation to nongravitational forces
acting on the satellites [Vigue et al., 1992]. For this reason
CM coordinate solutions are often transformed so that there
is no-net translation with respect to some previously established CM frame, which may require site ties (local surveys)
to collocated stations [Blewitt et al., 1992; Altamimi et al.,
2001]. VLBI site velocities have been placed in the CM
frame by applying a global velocity that minimizes velocity
differences with SLR at sites collocated by the two techniques [Watkins et al., 1994]. In all these cases that involve
frame transformations, however, it must be recognized that
the degree to which the solution is useful as a CM solution
depends on the accuracy and the temporal resolution of the
frame to which it is tied. For example, ITRF2000 [Altamimi
et al., 2001] might be useful as a CM frame to study secular
deformation but not for seasonal loading.
[29] To conserve momentum, redistribution of surface
mass displaces CE relative to CM by the geocenter displacement vector [tCE]CM, which satisfies [Trupin et al., 1992;
Dong et al., 1997; Blewitt et al., 2001]
½tCE CM ¼ ML rL =ME ¼ m=ME ¼ ½tCM CE ;
ð19Þ
which uses the definition of m from equation (9). From
equation (14), therefore, we have [aCM]CE = 1. Inserting
this into equation (17) and using Farrell’s numbers of
equation (18), we find
0
h1 CM ¼ h01 CE 1 ¼ 1:290;
0
l1 CM ¼ l10 CE 1 ¼ 0:887;
1 þ k10 CE ¼ 1 þ k10 CE 1 ¼ 0:
ð20Þ
[30] Note that the displacement field does not look like a
rigid body translation (h10 = l10 = 1), as has until now been
an unchallenged assumption in satellite geodetic analysis of
the geocenter. From equation (20), station coordinate displacements in the CM frame due to degree-one loading will
be 29% larger in the height component and 11% smaller in
lateral components than the displacements predicted by a
rigid body translation model. Even more significant is the
effect on relative position. For example, for a load moment
pointing along the z axis, the relative vector between a
station at the pole and a station on the equator changes
(along the z axis) by 40% of the rigid body translation
vector (also along the z axis).
[31] Although the total degree-one space potential has
zero effect on the satellite orbits in the CM frame, not
accounting for the stations displacements will affect the
determination of the satellite orbits, which will further
feedback into error in the determination of CM relative to
the stations in a complicated way. Therefore, methods that
estimate degree-one perturbations to satellite orbits (as
practiced in SLR) but do not allow the station network to
deform are inherently inconsistent. For example, Chen et al.
[1999] ignore the deformation, assuming that this effect is
‘‘very small (about 2%),’’ a number which correctly relates
to a frame origin effect [Dong et al., 1997] but which does
not account for relative station displacements (up to 40% of
the CE translation) and how this error propagates into the
estimated orbits.
4.3. Center of Surface Figure (CF)
[32] Let us define the center of surface figure, CF,
following the notation of Dong et al. [1997], who in turn
followed Trupin et al.’s [1992] definition of ‘‘center of
surface.’’ The CF frame is defined geometrically as though
the Earth’s surface were covered by a uniform, infinitely
dense array of points and the motions of these points are
taken into account. In practice, this can be realized by
appropriately averaging over a sufficiently dense global
distribution of geodetic stations, which presents no problem
in the case of GPS. In theory, the origin of the CF frame is
BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
such that the surface integral of the vector displacement
field is zero. This corresponds to no-net translation projected along any axis.
[33] It is important to recognize that rigid plate tectonics,
with boundaries that create and destroy surface, will generally produce a net translation of CF in the CM frame. The
average tectonic motion of the Earth’s surface is actually
more northward than southward, which implies a net
positive z component of secular translation of CF with
respect to the fixed point of plate rotation (modeled at the
center of a sphere). In contrast, physical arguments suggest
that CM cannot have significant secular motion with respect
to the center of the sphere [Argus et al., 1999, p. 29,080].
[34] It is therefore more appropriate to apply a no-net
translation condition to the residual station velocities
(observed minus plate rotation model), but strictly speaking,
this is not a CF frame. However, it is essential to calibrate
for tectonics when investigating loading. A residual CF
frame to study loading can be defined by applying a no-net
translation at every epoch with respect to a secular CM
frame [Davies and Blewitt, 2000]. Such a procedure has
been used to produce suitable CF frames for investigating
nonsecular loading [Van Dam et al., 1994, 2001; Blewitt et
al., 2001].
[35] In practice, the CF frame is well suited for techniques
where CM cannot be accurately realized. It is a natural
frame of choice for GPS, for which stochastic variation in
radiation pressure on the satellites is a limiting error source.
It is also a natural frame for VLBI using quasar sources,
with which no satellite dynamics are involved.
[36] To compute coordinate displacements in the CF
frame, we can integrate the vector surface displacements
in equation (16) and then solve for a such that the integral is
zero.
ZZ
½sCF d ¼ 0;
ZZ 0 ^ h d ¼ 0;
^ h þ h0
hm
l1 CF m hm
1 CF
(21)
ZZ 0
0
^ h d ¼ 0;
^ h þ h aCF hm
l1 aCF CE m hm
1
CE
ZZ 0
^ h.m
^
d ¼ 0:
l1 aCF CE m þ h01 l10 CE h
Given that m is a global constant not subject to surface
integration, we are free to take the dot product of the
integrand with m (to turn this into a scalar equation):
ZZ l10 aCF
CE
2 ^
d ¼ 0;
m2 þ h01 l10 CE h.m
ZZ 0
02
d ¼ 0;
l1 aCF CE þ h01 l10 CE Y10
(22)
4p 0
4p l10 aCF CE þ
h l10 CE ¼ 0;
3 1
1
½aCF CE ¼ h01 þ 2l10 CE ;
3
0
where Y10
is a degree-one zonal harmonic along the axis of
m. This result is consistent with that of Trupin et al. [1992]
ETG
10 - 7
(later used by Dong et al. [1997]), who integrated the
complex spherical harmonic form of loading deformation to
compute the translation between CF and CE. The difference
here is that I identify the parameter aCF with a load Love
number transformation, equation (17). Inserting this into
equation (17) and applying equation (18) gives
0
2
h1 CF ¼ h01 l10 CE ¼ 0:268;
3
0
1
l1 CF ¼ h01 l10 CE ¼ 0:134;
3
1
2
1 þ k10 CF ¼ 1 h01 l10
¼ 1:021:
3
3 CE
ð23Þ
[37] Comparing equations (23) and (18), we therefore
conclude that load Love numbers in the CE frame commonly used for dynamic modeling are not identical to those
in the CF frame commonly used in geodetic practice.
Fortunately the difference is small. Dong et al. [1997]
indicated that if the Earth were homogeneous, CF and CE
would be equivalent. For Farrell’s model, the trajectories of
CF and CE agree to within 2% of their magnitudes. Therefore, an advantage of CF is that it approximates the center of
mass of the solid Earth.
4.4. Center of Surface Lateral Figure (CL)
[38] In theory, the center of lateral figure (CL) frame is
such that the surface integral of the horizontal vector
displacement field is zero. In practice this type of constraint
has been applied to residuals of observed tectonic motions
minus modeled plate tectonic motions [Heki, 1996; Ma and
Ryan, 1997]. Such a method serves to realize a frame
consistent with plate tectonics but does not realize a true
CL frame because plate tectonics produces a nonzero net
horizontal vector displacement (again, due to surface creation and destruction). However, such a procedure can
effectively produce a ‘‘residual CL frame’’ for purposes of
investigating loading dynamics. Lavallée [2000] defines a
type of residual CL frame that minimizes the rate of
spherical distance between stations on rigid plate interiors,
without use of an a priori plate rotation model.
[39] The CL reference frame can be useful if vertical
displacements are much less accurately determined (or modeled) than are lateral displacements. The CL frame may be
applicable to radio techniques such as GPS and VLBI, where
vertical accuracy can be limited by errors in atmospheric
refractivity due to stochastic water vapor variations.
[40] By inspection of equation (17), choosing aCL = l10
defines a reference frame in which there is no lateral motion
caused by the degree-one deformations. Since only the
degree-one deformations can produce average translations
of the Earth’s surface, we can conclude that this defines the
CL frame.
0
h1 CL ¼ h01 l10 CE ¼ 0:403;
0
l1 CL ¼ l10 l10 CE ¼ 0;
0
k1 þ 1 CL ¼ 1 l10 CE ¼ 0:887:
ð24Þ
[41] Note that, in comparison with the CE frame, commonly used for dynamic modeling, degree-one vertical
ETG
10 - 8
BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
deformations in the CL frame are enhanced by 30%. This
type of effect might occur if stations are constrained to
move with constant horizontal velocities.
4.5. Center of Surface Height Figure (CH)
[42] The no-net height displacement frame (CH) is a
center of figure frame such that the vector average of height
displacements over the Earth’s surface is zero. This is
perhaps the most intuitive geometrical frame, as CH is
centered on the spherical shape of the Earth’s surface, which
remains spherical under degree-one deformation. CH is a
natural frame for models of vertical motion only, where the
globally averaged vertical motion is believed to be better
modeled than is horizontal motion or where geological plate
rotation models are suspect [Gerasimenko and Kato, 2000].
In practice, this type of constraint has been applied to
velocity residuals of observed rebound minus modeled
post-glacial rebound [Argus, 1996], thus realizing a ‘‘residual (rebound-adjusted) CH frame.’’ Vertical velocities of
several stations in global solutions have been constrained to
zero (C. Ma et al., Goddard Space Flight Center (GSFC)
VLBI solutions GLB886a; GLB907, 1993, 1994, available
at http://lupus.gsfc.nasa.gov/global/glb.html), thus effectively realizing a type of CH frame (though not in a true
global-average sense).
[43] By inspection of equation (17), choosing aCH = h10
defines a reference frame in which there is no height motion
caused by the degree-one deformations. Since only the
degree-one deformations can produce average translations
of the Earth’s surface, we can conclude that this defines the
CH frame.
0
h1 CH ¼ h01 h01 CE ¼ 0;
0
l1 CH ¼ l10 h01 CE ¼ 0:403;
1 þ k10 CH ¼ 1 h01 CE ¼ 1:290:
ð25Þ
[44] Note that, in comparison with the CE frame commonly used for dynamic modeling, degree-one lateral deformations are enhanced by a factor of 3 and are now even
larger than the height deformations in the CE frame. This
type of effect might occur if stations are constrained to have
no height motion. It is clear that horizontal motion can
entirely absorb degree-one deformations; therefore, the danger of constraining the heights of stations around the globe is
that it can greatly magnify loading variations in the horizontal. Also note that the total potential at radius RE in the
CH frame is precisely equal to the total body potential at the
Earth’s surface given by equation (6), which of course it
should, given that in the CH frame, the surface remains at
radius RE.
4.6. General Isomorphic Frames
[45] In principle, other isomorphic frames can be defined
by constraining any of the degree-one load Love numbers,
or linear combinations thereof, to any value. Isomorphic
frames can also be generated by taking linear combinations
of parameter a associated with other isomorphic frames. In
these cases it is straightforward to apply the analysis method
here to derive new load Love numbers. Whether or not
combination frames are practical depends on the problem at
hand. For example, Argus et al. [1999] produce a combi-
nation of a rebound-adjusted CH frame together with a
residual CL frame (using rebound-adjusted velocities with
respect to estimated rigid plate motions).
[46] Frames that constrain the coordinates of individual
stations will generally not satisfy equation (16) and so are
not isomorphic. Even minimal constraint frames will generally not be isomorphic if constrained station coordinates
do not incorporate an accurate loading displacement model.
Frames that are not isomorphic should be avoided, as
interpretation would not be straightforward. Fiducial-free
analyses, however, allow for nonrigid body deformation.
Coordinate time series can be constructed for station displacements relative to linear combinations of station displacements that (ideally) would satisfy the definition of a
convenient isomorphic frame [Blewitt et al., 2001]. Interpretation could then proceed on the understanding that we
must use the degree-one load Love number appropriate to
the chosen reference frame.
4.7. Geocenter Variations
[47] A change in the center of mass of surface loads
induces a detectable translation of the solid Earth relative to
the center of satellite orbits [Watkins and Eanes, 1997; Chen
et al., 1999]. Observations of this phenomenon help to
constrain models involving global redistribution of mass,
especially the water cycle and how it might be affected by
global climate change [Chao et al., 1987; Trupin et al.,
1992; Dong et al., 1997]. The physical principle behind the
‘‘geocenter’’ phenomenon is conservation of momentum,
and the only property of the solid Earth that is relevant is its
mass.
[48] However, it is important to recognize that the real
distance-changing deformation associated with change in
the center of mass of the surface load is not a negligible
effect, despite the (perhaps misleading) fact that the CF
frame closely approximates the CE frame. As seen previously, the vectors between individual stations vary by up to
40% of the geocenter trajectory. Hence, a cause for concern
is that previous methods determining the geocenter trajectory might be biased unless they also account for the sizable
distance-changing deformation associated with change in
load center of mass.
[49] The term ‘‘geocenter’’ generically refers to the translation of the origin between terrestrial reference frames;
however, its definition is not universally accepted. For
example, Chen et al. [1999] define the geocenter as the
translation of CM with respect to CE; Dong et al. [1997]
define the geocenter as the translation of CF with respect to
CM. Argus [1996] defines the geocenter as the translation of
rebound-adjusted CH (with the intent of aligning to the
rebound model’s CE frame) with respect to CM. Argus et al.
[1999] define the geocenter generally as any geometrical
center of figure with respect to CM.
[50] Table 1 summarizes the results of our derivations for
load Love numbers in various isomorphic frames appropriate for geodetic analysis, along with the translations of
origin. I choose to express isomorphic parameters relative to
CE because that is an appropriate frame for solid Earth
dynamics. However, I choose the convention of Argus et al.
[1999], expressing translations with respect to CM, because
CM is unperturbed by load redistribution and is the natural
reference frame origin for modeling satellite dynamics.
ETG
BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
10 - 9
Table 1. Degree-One Load Love Numbers and Relative Translations for Various Isomorphic Frames
Load Love Number
Description of Frame B
Center
Center
Center
Center
Center
of
of
of
of
of
mass of solid Earth, CE
mass of entire Earth, CM
surface figure, CF
lateral surface figure, CL
height surface figure, CH
Isomorphic Parameter
Relative to CE [aB]CE
Height
[h10 ]B
Lateral
[l10 ]B
Space Potential
[1 + k10 ]B
Body Potential
[1 + k10 h10 ]B
Geocenter Relative
to CM [tB]CM
0
1
0.290
1.290
0.269
0.403
0
0.113
0.887
0.134
0
0.403
1
0
1.021
0.887
1.290a
1.290a
1.290a
1.290a
1.290a
1.290a
m/ME
0
1.021(m/ME)
0.887(m/ME)
1.290(m/ME)
1 0
3[h1
+ 2l10 ]CE = 0.021
[l10 ]CE = 0.113
[h10 ]CE = 0.290
a
This provides a consistency check, knowing that the body potential calculated at a point moving with the Earth’s surface should not depend on the
choice of origin.
[51] The translation of origin of an isomorphic frame B as
viewed in the CM frame is related to the load moment by
the following formula:
½tB CM ¼ ½tB tCM CE ¼ ½aB aCM CE m=ME ¼ ½aB CE 1 m=ME :
ð26Þ
Equation (26) was used to generate the location of CM in
the various frames of Figure 2. It has been previously noted
that, in inertial space, the trajectory of the CF frame
approximates the CE frame to within 2% [Dong et al.,
1997]. Here I have shown that center of figure frames
defined by no-net vector translation using only lateral
displacements CL or only height displacements CH have
trajectories that differ from that of CF by 13– 27%.
[52] Geocenter differences (between frame types) have
been noted experimentally. Argus et al. [1999] found
secular geocenter variation at the 2 – 4 mm/yr level in
differences of solutions in the CH, CL, and CM frames
derived by Watkins et al. [1994], Heki [1996], Argus [1996],
and C. Ma et al. (various GSFC VLBI solutions, 1993 –
1996). While some of these differences might be explained
in terms of systematic error, to date there has been little
theoretical explanation as to how real differences can arise
from dynamic processes. The theory developed here at least
provides a predictive basis for computing station coordinate
variations that result from elastic crustal loading, using an
approach that ought to be more broadly applicable. Even
VLBI observations of quasars are indirectly sensitive to the
geocenter associated with elastic loading, as the degree-one
deformation that must accompany geocenter displacement
changes the distances between radio telescopes [Lavallée
and Blewitt, 2002].
[53] For completeness, note that the deformation relative
to the inertial origin CM can always be constructed as the
superposition of the deformation expressed in any frame B
plus that frame’s geocenter motion. Applying equation (15),
^ ½tB ½sh CM ¼ ½sh B þh.
CM
½sl CM ¼ ½sl B þ^l. ½tB CM
ð27Þ
^ ½t B :
½tB CM ½U CM ¼ ½U B þgh.
CM
5. Conclusions
[54] Good geodetic practice and subsequent scientific
interpretation requires self-consistency in dynamic models
and terrestrial reference frame theory. This especially applies
when attempting to separate global-scale, low-frequency
processes and processes that involve degree-one deformation associated with surface load redistribution. Here I have
identified the class of ‘‘isomorphic frames,’’ which have the
property that, under redistribution of the load characterized
by load moment m, the origins translate relative to each other
along the axis of m, and the deformation dynamics can be
formulated using frame-dependent degree-one load Love
numbers. The following frames have been shown to be
isomorphic: center of mass of the solid Earth, center of mass
of the entire Earth system, no-net translation of the surface,
no-net horizontal translation of the surface, and no-net
vertical translation of the surface. Degree-one load Love
numbers in the CE frame commonly used for dynamic
modeling are strictly not applicable to frames used in geodetic practice. The degree-one load Love number transformations and geocenter translations can be characterized
by a single isomorphic parameter a, whose value can be
derived, given the definition of a conceptual reference frame.
[55] As far as degree-one deformations are concerned,
there is no fundamental distinction between the concepts of
vertical motion versus horizontal motion, as they are shown
to depend entirely on the choice of frame. Therefore, great
care must be exercised when interpreting global-scale
redistribution of surface loads, especially when the research
question involves the concepts of vertical and horizontal.
Examples demonstrate that peak surface displacements can
occur where the magnitude of the load distribution is
minimum, illustrating that local models of loading in the
IERS conventions [McCarthy, 1996] are misleading for the
investigation of global-scale phenomena.
[56] The degree-one displacement field does not look like
a rigid body translation. Previous methods for determining
the geocenter trajectory might be biased unless they also
account for the sizable distance-changing deformation associated with change in the load center of mass. Errors in the
rigid body translation model amount to 30% at specific
sites and up to 40% in relative vectors. When using a
geometrically-defined origin (CH) that minimizes height
displacements, lateral deformations can be magnified by a
factor of 3 over those predicted in the CE frame, typically
used by modelers, which is even larger than the height
deformations in the CE frame. This type of effect might
occur if a frame is realized by constraining the height of
some stations around the globe. I therefore recommend that
fiducial-free analysis be applied in geodesy, using this to
construct coordinate time series in an appropriate realization
of an isomorphic frame, with subsequent interpretation (or
model inversion) using degree-one load Love numbers
appropriate to the selected frame.
ETG
10 - 10
BLEWITT: SELF-CONSISTENCY IN REFERENCE FRAMES
[57] Geocenter differences (between frame types) have
been investigated experimentally, but predictive theory has
been lacking. The theoretical development presented here
quantitatively predicts differences in geocenter trajectories
in different frames if there is a global-scale redistribution of
surface mass. Differences in geocenter estimates, however,
will generally also depend on solid Earth processes other
than elastic loading, such as postglacial isostatic rebound,
and they will depend on how well ideal reference frames
have been realized by (imperfect) distributions of geodetic
stations. Further development is needed along the lines of
this research to better define a frame for studies of global
climate change, for example, a frame to improve interpretation of global sea level rise. In particular, more theoretical
development with the requisite clarity of concepts should in
future clarify how, in principle, one can separate secular
tectonics signals from decadal-scale loading, using decadalscale spans of space geodetic data.
Appendix A
[58] The following identity was used to derive equation
(13). The proof is simplified by choosing a spherical
coordinate system (latitude j, longitude l) such that the
polar axis points in the direction of m. By simple geometry
^ @f þ
and the definition of the surface gradient r ¼ j
^ ð1 cos fÞ@l , we can deduce
l
^ ð1= cos fÞ@l ðm sin fÞ
^l. r h.m
^
^ @f þ l
¼ ^l. j
^ m cos f þ 0 ¼ ^l. j
^ ðm. j
^ Þ:
¼ ^l. j
(A1)
But we can also write the following, remembering that our
choice of polar axis requires that ml = 0 everywhere:
^ l þ hm
^l.m ¼ ^l. j
^ h ¼ ^l. j
^ mf þ lm
^ ðm. j
^ Þ þ 0 þ 0;
ðA2Þ
hence proving the equivalence.
[59] Acknowledgments. I am grateful to P. Clarke and D. Lavallée
for useful discussions while developing this paper. I also thank B. Chao, D.
Argus, and K. Heki for helpful suggestions that I used to improve the final
version. This work was supported by the Geophysics Program, National
Science Foundation, grant EAR-0125575.
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G. Blewitt, University of Nevada, Nevada Bureau of Mines and Geology,
Mail Stop 178, Reno, NV 89557, USA. ([email protected])